Questions: Are these pairs of quantities proportional to each other? For the quantities that are proportional, what is the constant of proportionality? a. Radius and diameter of a circle b. Radius and circumference of a circle c. Radius and area of a circle d. Diameter and circumference of a circle e. Diameter and area of a circle

Are these pairs of quantities proportional to each other? For the quantities that are proportional, what is the constant of proportionality? a. Radius and diameter of a circle b. Radius and circumference of a circle c. Radius and area of a circle d. Diameter and circumference of a circle e. Diameter and area of a circle
Transcript text: Are these pairs of quantities proportional to each other? For the quantities that are proportional, what is the constant of proportionality? a. Radius and diameter of a circle b. Radius and circumference of a circle c. Radius and area of a circle d. Diameter and circumference of a circle e. Diameter and area of a circle
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Solution

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Solution Steps

Step 1: Determine if the Radius and Diameter of a Circle are Proportional

The diameter dd of a circle is twice the radius rr, i.e., d=2rd = 2r. This relationship is linear, and the two quantities are proportional. The constant of proportionality is 2.

Step 2: Determine if the Radius and Circumference of a Circle are Proportional

The circumference CC of a circle is given by C=2πrC = 2\pi r. This is a linear relationship, indicating that the radius and circumference are proportional. The constant of proportionality is 2π2\pi.

Step 3: Determine if the Radius and Area of a Circle are Proportional

The area AA of a circle is given by A=πr2A = \pi r^2. This is a quadratic relationship, not linear, so the radius and area are not proportional.

Final Answer

  • a. Radius and diameter of a circle: Proportional, constant of proportionality is 2\boxed{2}.
  • b. Radius and circumference of a circle: Proportional, constant of proportionality is 2π\boxed{2\pi}.
  • c. Radius and area of a circle: Not proportional.
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